## Abstract

We present a study of Rainich-like conditions for symmetric and trace-free tensors *T*. For arbitrary even rank we find a necessary and sufficient differential condition for a tensor to satisfy the source-free field equation. For rank 4, in a generic case, we combine these conditions with previously obtained algebraic conditions to gain a complete set of algebraic and differential conditions on *T* for it to be a superenergy tensor of a Weyl candidate tensor, satisfying the Bianchi vacuum equations. By a result of Bell and Szekeres, this implies that in vacuum, generically, *T* must be the Bel–Robinson tensor of the spacetime. For the rank 3 case, we derive a complete set of necessary algebraic and differential conditions for *T* to be the superenergy tensor of a massless spin-3/2 field, satisfying the source-free field equation.

## 1. Introduction

Given a symmetric trace-free divergence-free tensor *T*_{ab} satisfying the dominant energy condition (*T*_{ab}*u*^{a}*v*^{b}≥0 for all future-directed causal vectors *u*^{a} and *v*^{a}), one can ask what more is required of *T*_{ab} for it to be the energy–momentum tensor of some given physical field. To characterize *T*_{ab} completely, it turns out that we will need both an algebraic condition and a differential condition. Assuming dimension 4 and Lorentzian metric, the following is a result in classical Rainich–Misner–Wheeler theory (Rainich 1925; Misner & Wheeler 1957).

*A symmetric trace-free tensor T*_{ab} *that satisfies the dominant energy condition can be written as:* *, where F*_{ab} *is a 2-form,* *if and only if*(1.1)

Here, ^{*}*F*_{ab} is the dual 2-form of *F*_{ab}. Removing the assumption of the dominant energy condition, theorem 1.1 is still true up to sign (Bergqvist & Senovilla 2001): if and only if equation (1.1) is satisfied.

A tensor *T*_{ab} fulfilling the requirements of the theorem is algebraically the energy–momentum tensor of a Maxwell field *F*_{ab}. Equivalently, a tensor satisfying the given requirements can be written as , where *φ*_{AB} is a spinor representing the Maxwell field. In the theorem, *F*_{ab} is only determined up to a duality rotation , which corresponds to .

Of course, this algebraic condition will have to be accompanied by a differential condition that assures that the field *F*_{ab} (or equally, *φ*_{AB}) satisfies the source-free Maxwell's equations. The following is known (Rainich 1925; Misner & Wheeler 1957).

*Suppose that* *for some 2-form* *and that* *. Then,* *for some 2-form F*_{ab} *satisfying the source-free Maxwell equations* *if and only if*(1.2)

Note that *F*_{ab} is obtained from by a duality rotation and that the source-free Maxwell equations in spinor form are simply (Penrose & Rindler 1984). Using spinors, theorem 1.2 can also be written as follows.

*Suppose that* *for some symmetric spinor ϕ*_{AB} *and that* *. Then,* *for some symmetric spinor φ*_{AB} *satisfying* *if and only if equation* *(1.2)* *is satisfied*.

The validity of theorems 1.2 and 1.3 is obviously restricted to cases where *T*_{ab}*T*^{ab}≠0, i.e. when the two principal null directions of *φ*_{AB} are different (non-null electromagnetic fields). In the null case, results cannot be stated in an equally simple way (see Geroch 1966; Ludwig 1970).

Theorems 1.1 and 1.2 imply the following.

*A symmetric trace-free and divergence-free tensor T*_{ab} *with T*_{ab}*T*^{ab}≠0 *is, up to sign, the energy–momentum tensor of a source-free Maxwell field if and only if equations* (1.1) and (1.2) *are satisfied*.

If one uses Einstein's equation, *T*_{ab} may be replaced by the Ricci tensor *R*_{ab} in the corollary since *T*_{ab} is trace-free. In this case, the Ricci tensor is automatically divergence-free so this is not needed as a condition. Equations (1.1) and (1.2) are then satisfied for *R*_{ab} if and only if *R*_{ab} is the Ricci tensor for an Einstein–Maxwell spacetime.

The algebraic result of theorem 1.1 has been generalized to an arbitrary dimension and an arbitrary trace of *T*_{ab} when equation (1.1) is assumed (Bergqvist & Senovilla 2001), and to cases in higher dimension when equation (1.1) is replaced by a third-order equation for *T*_{ab} (Bergqvist & Höglund 2002). In these generalizations, only rank 2 tensors *T*_{ab} were considered. Here, we will generalize theorems 1.1 and 1.2 to include symmetric trace-free tensors of ranks 3 and 4. For higher rank tensors, the dominant energy condition is replaced by a generalization called the *dominant property*,(1.3)for all causal vectors . The spacetime dimension will always be four here and the metric will be assumed to be of Lorentzian signature. The methods will be spinorial, and we will begin by reviewing necessary facts about these. After that, a differential condition for symmetric trace-free and divergence-free tensors of even rank is obtained, generalizing theorem 1.2, and the condition is applied to the Bel–Robinson tensor. The algebraic condition for the Bel–Robinson tensor was already obtained in Bergqvist & Lankinen (2004) and we can now give a complete characterization of the Bel–Robinson tensor. The Bel–Robinson tensor is the so-called superenergy tensor of the Weyl tensor or the Weyl spinor. To any tensor on a Lorentzian manifold, there is a corresponding superenergy tensor of even rank and this always has the dominant property (Bergqvist 1999; Senovilla 2000). In Lankinen & Pozo (2004), this definition was extended to include also superenergy tensors of spinors, which may then be of odd rank. Here, we derive both algebraic and differential conditions on symmetric trace-free and divergence-free tensors of rank 3, giving a complete characterization of superenergy tensors of massless spin- fields.

## 2. Some useful spinor identities

We review some well-known facts about spinors that will be important to us. The formulae can be found in Penrose & Rindler (1984) and we also follow their notation and conventions (except for a factor 4 in the definition of the Bel–Robinson tensor). Spinor expressions for general superenergy tensors are given in Bergqvist (1999).

We use capital letters *A*, *B*, …, *A*′, *B*′, … for spinor indices and identify with tensor indices *a*, *b*, … according to *AA*′=*a*. A spinor , where represents some set of spinor indices, can be divided up into its symmetric and antisymmetric parts with respect to a pair of indices:The antisymmetric part can be written aswhere *ε*_{AB}=−*ε*_{BA}, so(2.1)From this one also has(2.2)A simple but very useful rule is(2.3)Note that if then we havewhere , so permuting *A* and *B* gives a trace reversal. From this, we find another formula we shall need (with not necessarily symmetric in *ab*)(2.4)The completely antisymmetric tensor *e*_{abcd}, normalized by *e*_{abcd}*e*^{abcd}=−24, can be written asRaising the indices *cd* and applying this tensor to the tensor gives the following useful relation:(2.5)For reference, we also state the relations between corresponding tensorial and spinorial objects of interest. The relation between a 2-form *F*_{ab} and a symmetric spinor *φ*_{AB} isand one also hasFor the Weyl tensor, *C*_{abcd}, and the completely symmetric Weyl spinor, *Ψ*_{ABCD}, the corresponding relations are(2.6)and(2.7)

That a tensor *T*_{a⋯b} is completely symmetric and trace-free is very elegantly expressed in an equivalent way using spinor indices asWe shall study when a tensor can be factorized in terms of spinors. If a tensor *τ*_{a⋯b} can be written as(2.8)for some spinor *Χ*_{A⋯B}, then it follows that *τ*_{a⋯b} satisfies the dominant property (1.3) and(2.9)Conversely, suppose that *τ*_{a⋯b} satisfies equation (2.9). Let *u*^{a}, …, *v*^{a} be future-directed null vectors such that *τ*_{a⋯b}*u*^{a}⋯*v*^{b}=*k*≠0. Such null vectors must exist because otherwise, by taking linear combinations, we would get *τ*_{a⋯b}*u*^{a}⋯*v*^{b}=0 for all vectors, which would imply *τ*_{a⋯b}=0. Then, write the null vectors in terms of spinors as . Contract equation (2.9) with these spinors to getfrom which it follows that *τ*_{a⋯b} and −*τ*_{a⋯b} can be factorized as in equation (2.8), one of them with and the other with an extra ‘i’ in the factor, and that either *τ*_{a⋯b} or −*τ*_{a⋯b} has the dominant property.

Finally, we introduce the following useful notation:for any tensor *T*_{a⋯b}.

## 3. Differential conditions for even rank

Suppose the tensor , with *r* even, can be factorized according towith symmetric. Then, is symmetric, trace-free and satisfies the dominant property. Note that is invariant under . We now prove a generalization of theorem 1.2 (or theorem 1.3).

*Let r be even and suppose that* *and* *for some totally symmetric* . *Then*, *for some totally symmetric* *satisfying* *if and only if*

Since is preserved under ‘rotations’ (*Χ* real), we may assume thatis real (otherwise rotate with a suitable *Χ*).

Now, we want to find the condition for the existence of some with , and . Clearly, we can write for some real *θ* with as above. If satisfies the given field equations, then we have (using the Leibniz rule)Cancelling the e^{−iθ} and contracting with , we getUsing equations (2.2) and (2.3) and the fact that *r* is even we haveso we arrive atRelabelling *A*_{1} and *B* we getIf we define a vector as(3.1)then expanding by the Leibniz rule and contracting by , we getorhence, the vector is purely imaginary and, therefore, *S*_{a} is a real vector.

We want to translate the right-hand side of equation (3.1) into a tensorial expression. Differentiate the tensor and make one contraction, leading toIf we contract this with we get, again assuming that *r* is even,Now we can use equation (3.1) to getOn the right-hand side, the first term is purely imaginary and the second is real. Thus, by taking the complex conjugate and then taking the difference, we getFinally, we can use equation (2.5) on the index pairs *A*_{1}*A*′_{1} and *BB*′ to getHere, 4*K*^{2}=*T*.*T* so we get the formula(3.2)Conversely, with *S*_{a} given by equation (3.2), there is a real solution *θ* (determined up to an additive constant) to the equation ∇_{a}*θ*=*S*_{a} if the integrability conditionis satisfied. This completes the proof. ▪

Note that the above proof does not hold for odd *r* in which case so *T*.*T*=0 as well.

## 4. Complete Rainich theory for the Bel–Robinson tensor for Petrov types *I*, *II* and *D*

As mentioned earlier, the algebraic Rainich condition for the Bel–Robinson tensor was obtained in Bergqvist & Lankinen (2004) but we restate the result here.

*A completely symmetric and trace-free rank 4 tensor T*_{abcd} *is, up to sign, a Bel–Robinson type tensor, i.e.* *, where C*_{abcd} *has the same algebraic symmetries as the Weyl tensor if and only if*(4.1)

Equivalently, this may also be stated as *T*_{abcd} is the superenergy tensor (Senovilla 2000) of a Weyl candidate tensor (that is, a tensor with same algebraic symmetries as the Weyl tensor: , , ). As shown in Bergqvist & Lankinen (2004) the identity in theorem 4.1 can be replaced byIn terms of spinors we can state theorem 4.1 as follows.

*A completely symmetric and trace-free rank 4 tensor T*_{abcd} *can be written as* *with Ψ*_{ABCD}=*Ψ*_{(ABCD)} *if and only if equation* *(4.1)* *is satisfied*.

Thus, from a spinorial viewpoint, this is a natural generalization of the classical Rainich theory. We can then ask the same question as in the classical case, e.g. what is required in order to have *C*_{abcd} (or *Ψ*_{ABCD}) satisfy some field equations? In this case, we choose the source-free gravitational field equation(4.2)for the Weyl spinor that holds whenever Einstein's vacuum equations hold. The tensor form of equation (4.2) is the vacuum Bianchi identity ∇_{[a}*C*_{bc]de}=0 (⇔∇^{a}*C*_{abcd}=0 in four dimensions) for the Weyl tensor. From theorem 3.1, we immediately have the following generalization of theorem 1.2.

*If* *for a completely symmetric spinor Φ*_{ABCD} *and if* ∇^{a}*T*_{abcd}=0*, then in a region where T*.*T*≠0 *we have* *for a completely symmetric spinor Ψ*_{ABCD} *satisfying* ∇^{AA′}*Ψ*_{ABCD}=0 *if and only if*(4.3)

This corollary gives a differential Rainich-like condition on the Bel–Robinson tensor. Combining theorem 4.1 (or theorem 4.2) and corollary 4.3 we get the rank 4 generalization of corollary 1.4, which gives the complete Rainich theory for Bel–Robinson type tensors. The tensor version is as follows.

*Suppose that T*_{abcd} *is completely symmetric, trace-free and divergence-free and that T*.*T*≠0. *Then* *for a Weyl candidate tensor C*_{abcd} *satisfying* ∇_{[a}*C*_{bc]de}=0 *if and only if equations* (4.1) and (4.3) *are satisfied*.

Expressed in terms of spinors we get the following.

*Suppose that T*_{abcd} *is completely symmetric, trace-free and divergence-free and that T*.*T*≠0. *Then* *for a completely symmetric spinor Ψ*_{ABCD} *satisfying* ∇^{AA′}*Ψ*_{ABCD}=0 *if and only if equations* (4.1) and (4.3) *are satisfied*.

We now proceed to see when these conditions imply that *C*_{abcd} is not only a Weyl *candidate* tensor satisfying ∇_{[a}*C*_{bc]de}=0, but the actual Weyl tensor of the spacetime. Firstly, *T*.*T*=0 if and only if the spacetime is of Petrov type *III* or *N* (Bergqvist 1998). Thus, we restrict ourselves to spacetimes of Petrov type *I*, *II* and *D*. Bell & Szekeres (1972) call a spacetime in which equation (4.2) is satisfied by the actual Weyl spinor a *C-space*; hence all vacuum spacetimes are *C*-spaces. This is also equivalent to the vanishing of the Cotton tensor (García *et al*. 2004). For spacetimes of Petrov type *I*, Bell and Szekeres prove the following:

*In an algebraically general C-space, the source-free field equations* *(4.2)* *(the vacuum Bianchi identities) have a unique solution to within constant multiples*, *or its solutions are linear combinations of two independent solutions at most*.

In Bell & Szekeres (1972), conditions for the cases with non-unique solutions are given and the authors claim that most physically acceptable metrics do not satisfy these conditions. As the conditions are not so simply stated, we refer the reader to Bell & Szekeres (1972) for further discussion. With the exception of these cases, there is, up to a multiplicative constant, a unique solution to equation (4.2), which is then, of course, the Weyl spinor (so the gravitational field is uniquely determined by the Bianchi identities).

For Petrov types *II* and *D*, let *o*_{A}, *ι*_{A} be a spin basis such that *o*_{A} is the repeated principal null direction in spacetimes of Petrov type *II*, and such that *o*_{A} and *ι*_{A} are the repeated principal null directions in spacetimes of Petrov type *D*, and for the remaining of this section, let *Ψ*_{ABCD} denote the actual Weyl spinor of spacetime. Then, the following was proved in Bell & Szekeres (1972).

*In a C-space of Petrov type II, the solution Φ*_{ABCD} *of the source-free field equations* *(4.2)* *is unique up to a constant α and null type fields* *with β a scalar, according to* *, where Ψ*_{ABCD} *is the Weyl spinor*. *For Petrov type D the solution can be written as * *, where* *with γ a scalar*.

In deriving these theorems, Bell & Szekeres use the Buchdahl conditions (Penrose & Rindler 1984):which are algebraic consistency conditions that relate any solution *Φ*_{A1⋯An} of the spin- equation to the Weyl spinor *Ψ*_{ABCD}. Using these results, we have the following.

*In C-spaces (including vacuum spacetimes), if T*_{abcd} *is completely symmetric, trace-free and divergence-free, then, generically (Petrov type I and excluding the exceptions given in* *Bell & Szekeres 1972**) and up to a constant factor, T*_{abcd} *is the Bel–Robinson tensor of spacetime if and only if equations* (4.1) and (4.3) *are satisfied*.

For Petrov types *II* and *D*, the following weaker conclusion can be drawn.

*In C-spaces (including vacuum spacetimes), if T*_{abcd} *is completely symmetric, trace-free and divergence-free and if spacetime is of Petrov type II (D), then* *, where* *if and only if equations* (4.1) and (4.3) *are satisfied*.

Note that the freedom in these cases does not preserve the principal null directions or even the Petrov type.

## 5. Algebraic conditions for rank 3

In Senovilla's original definition of superenergy tensors of arbitrary tensors (Senovilla 2000), all superenergy tensors are of even rank. However, in Bergqvist & Senovilla (1999), tensors of the form were used to study causal propagation of spin- fields. In Lankinen & Pozo (2004), Senovilla's definition was extended to include superenergy tensors of spinors and these may be of odd rank. Then, for instance, the superenergy tensor of a completely symmetric spinor of arbitrary rank is . We now study Rainich type conditions for the rank 3 case, beginning with an algebraic characterization.

*A completely symmetric and trace-free rank 3 tensor T*_{abc} *can be written as* *with Ψ*_{ABC}=*Ψ*_{(ABC)} *if and only if*(5.1)

From the results in §2, we must prove that equation (5.1) is equivalent to(5.2)We follow the method developed by Bergqvist & Lankinen (2004) and divide up the left-hand side in symmetric and antisymmetric parts with respect to the pairs *A*′*D*′, *B*′*E*′ and *C*′*F*′. Antisymmetric parts correspond to traces so for terms with 3, 2, 1 or 0 symmetrizations we have, respectively,Therefore, equation (5.2) is equivalent toNow, continuing in the same way with respect to the unprimed indices of these two expressions, expressions with an odd total number of contractions vanish. Hence equation (5.2) is equivalent to(5.3)Now, divide into symmetric and antisymmetric parts four times in the index pairs *A*′*D*′, *AD*, *BE* and *B*′*E*′. Again, terms with an odd number of contractions vanish and we getSince an expression is zero if and only if all its symmetric and antisymmetric parts are zero, then equation(5.2) is equivalent to(5.4)Now, note that by using equation (2.2) on *A*′*B*′, we getApplying equation (2.2) with respect to *DE* in the first term and *AE* in the second we haveIn the same way, first acting on *DE* and then on *A*′*B*′ and *B*′*D*′, we findSubstituting these expressions into equation (5.4) givesLowering indices, we use equation (2.4) to rewrite this as(5.5)where we also used *T*_{jkl}*T*^{jkl}=0. Since equation (5.5) is equivalent to equation (5.2), the proof is now complete. ▪

## 6. Differential conditions for rank 3

It is clear that the methods of §3 do not work for odd rank. We have for example that *T*_{a⋯b}*T*^{a⋯b}=0 in this case and it is important whether equation (2.3) is used an even or an odd number of times. We present here a condition for rank 3 but it can be generalized to higher odd rank. Given a completely symmetric spinor *Ψ*_{ABC}, we define a symmetric spinor asWritingwhere *α*_{A}, *β*_{A} and *γ*_{A} are the three principal null directions of *Ψ*_{ABC}, we may say that *Ψ*_{ABC} is of type *I*, *II* or *N* if the principal null directions are all distinct, if two coincide, or if all three coincide, respectively. It is then easy to see that *ψ*_{AB}=0 if and only if *Ψ*_{ABC} is of type N and that *ψ*_{AB}*ψ*^{AB}≠0 if and only if *Ψ*_{ABC} is of type *I*. With we see if and only if *Ψ*_{ABC} is of type *I*. For type *I*, the generic case, we have the following.

*Suppose that* *for some symmetric spinor Φ*_{ABC} *and that* . *Then,* *for some symmetric spinor Ψ*_{ABC} *satisfying* ∇^{AA′}*Ψ*_{ABC}=0 *if and only if*(6.1)

Since is preserved under ‘rotations’ (*Χ* real), we may assume that the symmetric spinor has the property thatis real (otherwise rotate with a suitable *Χ*). Now, we want to find the condition for the existence of some *Ψ*_{ABC} with *Ψ*_{ABC}=*Ψ*_{(ABC)}, and ∇^{AA′}*Ψ*_{ABC}=0. Clearly, we can write for some real *θ*. The differential equation becomesMultiplying by , we haveThen, multiply by *ϕ*_{DE} and use that is antisymmetric in *AE*. This implies thatHence,Define a vector(6.2)which is real since by applying the Leibniz rule to ; contracting with and using , one finds that the vector is purely imaginary.

Next, translate the right-hand side of equation (6.2) into a tensorial expression. We haveSubtract the complex conjugate to getand apply equation (2.5) to get(6.3)Conversely, with the real vector *S*_{a} given by equation (6.3), the equation ∇_{a}*θ*=*S*_{a} has a real solution *θ* (determined up to an additive constant) if the integrability conditionis satisfied. This proves the theorem. ▪

## 7. Complete Rainich theory for rank 3

A symmetric rank 3 spinor *Ψ*_{ABC} can be seen as representing a spin- field on spacetime. The field equations for a massless spin- field arewhich are of the form in theorem 6.1 above. Thus, collecting together the algebraic and differential conditions for symmetric trace-free and divergence-free rank 3 tensors obtained above, we find the following.

*Suppose that T*_{abc} *is symmetric, trace-free and divergence-free and that* . *Then T*_{abc} *is the superenergy tensor of a massless spin*- *field, i.e.* *for some symmetric spinor Ψ*_{ABC} *satisfying* *, if and only if**and*

In analogy with the rank 2 and rank 4 cases, this can be seen as a complete Rainich theory, in the mathematical sense since *T*_{abc} is not linked directly to the geometry via the field equations in present physical theories, for rank 3 superenergy tensors for the generic (type I) case.

## 8. Discussion

We have presented a complete Rainich theory for superenergy tensors of rank 3 and 4 in four dimensions in a generic case. However, the results obtained may be generalized to higher rank superenergy tensors. The interpretation is clear since the equations involved are the equations for a massless spin- field. It is also possible to make other generalizations of these results. For example, one could consider massive spin- fields, in which case it is obviously necessary to modify theorems 3.1 and 6.1. One could also consider the rank 4 differential conditions in spacetimes of Petrov type *III* and *N*, where *T*.*T*=0 and theorem 3.1 does not apply. From the results for the rank 2 case (Geroch 1966; Ludwig 1970), it is likely that this case will be rather complicated and that it is not so easy to apply Bell–Szekeres (Bell & Szekeres 1972) types of results here (which are already complicated for any algebraically special case). Note, however, that the algebraic conditions also apply to cases when *T*.*T*=0. For generalizations to arbitrary spacetime dimension or to metrics of arbitrary signature, tensor methods would be needed and it is clear that these would be much more complicated than the spinor methods we have used here.

## Acknowledgements

We thank Brian Edgar, José Senovilla and a referee for useful suggestions and comments.

## Footnotes

- Received May 3, 2004.
- Accepted September 14, 2004.

- © 2005 The Royal Society